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```SUBROUTINE GELS

PURPOSE
TO SOLVE A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS WITH
SYMMETRIC COEFFICIENT MATRIX UPPER TRIANGULAR PART OF WHICH
IS ASSUMED TO BE STORED COLUMNWISE.

USAGE
CALL GELS(R,A,M,N,EPS,IER,AUX)

DESCRIPTION OF PARAMETERS
R	  - M BY N RIGHT HAND SIDE MATRIX.  (DESTROYED)
ON RETURN R CONTAINS THE SOLUTION OF THE EQUATIONS.
A	  - UPPER TRIANGULAR PART OF THE SYMMETRIC
M BY M COEFFICIENT MATRIX.	(DESTROYED)
M	  - THE NUMBER OF EQUATIONS IN THE SYSTEM.
N	  - THE NUMBER OF RIGHT HAND SIDE VECTORS.
EPS	  - AN INPUT CONSTANT WHICH IS USED AS RELATIVE
TOLERANCE FOR TEST ON LOSS OF SIGNIFICANCE.
IER	  - RESULTING ERROR PARAMETER CODED AS FOLLOWS
IER=0  - NO ERROR,
IER=-1 - NO RESULT BECAUSE OF M LESS THAN 1 OR
PIVOT ELEMENT AT ANY ELIMINATION STEP
EQUAL TO 0,
IER=K  - WARNING DUE TO POSSIBLE LOSS OF SIGNIFI-
CANCE INDICATED AT ELIMINATION STEP K+1,
WHERE PIVOT ELEMENT WAS LESS THAN OR
EQUAL TO THE INTERNAL TOLERANCE EPS TIMES
ABSOLUTELY GREATEST MAIN DIAGONAL
ELEMENT OF MATRIX A.
AUX	  - AN AUXILIARY STORAGE ARRAY WITH DIMENSION M-1.

REMARKS
UPPER TRIANGULAR PART OF MATRIX A IS ASSUMED TO BE STORED
COLUMNWISE IN M*(M+1)/2 SUCCESSIVE STORAGE LOCATIONS, RIGHT
HAND SIDE MATRIX R COLUMNWISE IN N*M SUCCESSIVE STORAGE
LOCATIONS. ON RETURN SOLUTION MATRIX R IS STORED COLUMNWISE
TOO.
THE PROCEDURE GIVES RESULTS IF THE NUMBER OF EQUATIONS M IS
GREATER THAN 0 AND PIVOT ELEMENTS AT ALL ELIMINATION STEPS
ARE DIFFERENT FROM 0. HOWEVER WARNING IER=K - IF GIVEN -
INDICATES POSSIBLE LOSS OF SIGNIFICANCE. IN CASE OF A WELL
SCALED MATRIX A AND APPROPRIATE TOLERANCE EPS, IER=K MAY BE
INTERPRETED THAT MATRIX A HAS THE RANK K. NO WARNING IS
GIVEN IN CASE M=1.
ERROR PARAMETER IER=-1 DOES NOT NECESSARILY MEAN THAT
MATRIX A IS SINGULAR, AS ONLY MAIN DIAGONAL ELEMENTS
ARE USED AS PIVOT ELEMENTS. POSSIBLY SUBROUTINE GELG (WHICH
WORKS WITH TOTAL PIVOTING) WOULD BE ABLE TO FIND A SOLUTION.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
NONE

METHOD
SOLUTION IS DONE BY MEANS OF GAUSS-ELIMINATION WITH
PIVOTING IN MAIN DIAGONAL, IN ORDER TO PRESERVE
SYMMETRY IN REMAINING COEFFICIENT MATRICES.
```